Generalized Fourier-Feynman Transform and Sequential Transforms on Function Space

Abstract

In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional F given by F (x) =ν((e 1 , x) ∼ ,. .. , (en, x) ∼), where (e, x) ∼ denotes the Paley-Wiener-Zygmund stochastic integral with x in a very general function space C a,b [0, T ] andν is the Fourier transform of complex measure ν on B(R n) with finite total variation. We then define two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

L2-sequential transforms on function space

“…The sequential transforms on C a,b [0, T ] were introduced in [10,11]. In this section, we define L 2 -sequential function space transforms on C a,b [0, T ].…”

“…The stochastic process X on C a,b [0, T ] × [0, T ] defined by X(x, t) = x(t) is subject to a drift a(t) and is non-stationary in time. For more details, see [6][7][8][9][10][11][12]18,22].…”

Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral

“…On the other hand, in [5][6][7]9,14,15], the authors defined the generalized analytic Feynman integral and the generalized analytic Fourier-Feynman transform on the function space C a,b [0, T ], and studied their properties and related topics. The function space C a,b [0, T ], induced by a GBMP, was introduced by Yeh in [30], and was used extensively in [4,8,[10][11][12][13]20].…”

“…In this note, we set c = a(0) = b(0) = 0. Then the function space C a,b [0, T ] induced by the GBMP Y determined by the a(·) and b(·) can be considered as the space of continuous sample paths of Y , see [4][5][6][7][8][9][10][11][12][13][14][15][16]20], and one can see that for each t ∈ [0, T ],…”

“…However, the stochastic process used in this paper, as well as in [4][5][6][7][8][9][10][11][12][13][14][15][16]18,20,21,25,30], is non-stationary in time and is subject to a drift because In this note, we present an example of a bounded functional which is not generalized analytic Feynman integrable on the function space C a,b [0, T ].…”